Stochastic Dominance and Risk Measures

Abstract

There are various measures of risk (see Chap. 1) and each of them has its pros and cons. This chapter focuses on the notion of risk in the stochastic dominance framework. We first discuss the concept of mean preserving spread (MPS) and various risk measures suggested by Rothschild and Stiglitz (R&S) and then extend it to the case where riskless asset exists. Finally, we discuss the mean-preserving spread antispread (MPSA) which is a risk measure similar to the MPS but which corresponds to DARA risk-averse utility functions. Thus, the MPS corresponds to SSD and the MPSA corresponds to TSD. We deal in this chapter with theoretical measures of risk in the most general case where no constraints are imposed on the distributions of returns. In this framework the variance does not measure risk. However, when the distributions are Elliptic (the Elliptic family of distributions includes the normal, logistic and many other distributions) the variance is the precise measure of risk. Moreover, Markowitz and Levy and Markowitz have shown empirically that with no constraints on the distributions of return, selecting a portfolio from the MV efficient frontier almost precisely yields the same expected utility as selecting assets by a direct expected utility maximization. Thus, the variance is an excellent approximation to the precise measure of risk. However, in this chapter we define the mathematical precise measure of risk which is relevant not only to normal distributions or to empirical stock market data, but also to distributions with a very large asymmetry, e.g., to two lotteries the investor faces, two insurance policies, two options prospects etc.